PRINC

Computes principal components from a variance‑covariance matrix or a correlation matrix.

Required Arguments

NDF — Number of degrees of freedom in COV. (Input)
If NDF is less than or equal to 0, 100 degrees of freedom are assumed.

COV — NVAR by NVAR matrix containing the covariance or correlation matrix. (Input)
Only the upper triangular part of COV is referenced.

EVAL — Vector of length NVAR containing the eigenvalues from matrix COV ordered from largest to smallest. (Output)

Optional Arguments

NVAR — Order of matrix COV. (Input)
Default: NVAR = size (COV,2).

LDCOV — Leading dimension of COV exactly as specified in the dimension statement in the calling program. (Input)
Default: LDCOV = size (COV,1).

ICOV — Covariance/Correlation matrix option parameter. (Input)
ICOV = 0 means that a covariance matrix is input. Otherwise, a correlation matrix is input.
Default: ICOV = 0.

PCT — Vector of length NVAR containing the cumulative percent of the total variance explained by each principal component. (Output)

STD — Vector of length NVAR containing the estimated asymptotic standard errors of the eigenvalues. (Output)

EVEC — NVAR by NVAR matrix containing the eigenvectors of COV, stored columnwise. (Output)
Each vector is normalized to have Euclidean length equal to the value one. Also, the sign of each vector is set so that the largest component in magnitude (the first of the largest if there are ties) is made positive.

LDEVEC — Leading dimension of EVEC exactly as specified in the dimension statement in the calling program. (Input)
Default: LDEVEC = size( EVEC, 1).

A — NVAR by NVAR matrix containing the correlations of the principal components (the columns) with the observed/standardized variables (the rows). (Output)
If ICOV = 0, then the correlations are with the observed variables. Otherwise, the correlations are with the standardized (to a variance of 1.0) variables. In the principal component model for factor analysis, matrix A is the matrix of unrotated factor loadings.

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)
Default: LDA = size( A, 1).

FORTRAN 90 Interface

Generic: CALL PRINC (NDF, COV, EVAL [, …])

Specific: The specific interface names are S_PRINC and D_PRINC.

FORTRAN 77 Interface

Single: CALL PRINC (NDF, NVAR, COV, LDCOV, ICOV, EVAL, PCT, STD, EVEC, LDEVEC, A, LDA)

Double: The double precision name is DPRINC.

Description

Routine PRINC finds the principal components of a set of variables from a sample covariance or correlation matrix. The characteristic roots, characteristic vectors, standard errors for the characteristic roots, and the correlations of the principal component scores with the original variables are computed. Principal components obtained from correlation matrices are the same as principal components obtained from standardized (to unit variance) variables.

The principal component scores are the elements of the vector y = ΓTx where Γ is the matrix whose columns are the characteristic vectors (eigenvectors) of the sample covariance (or correlation) matrix and x is the vector of observed (or standardized) random variables. The variances of the principal component scores are the characteristic roots (eigenvalues) of the the covariance (correlation) matrix.

Asymptotic variances for the characteristic roots were first obtained by Girshick (1939) and are given more recently by Kendall, Stuart, and Ord (1983, page 331). These variances are computed either for covariance matrices (ICOV = 0) or for correlation matrices (ICOV ≠ 0).

The correlations of the principal components with the observed (or standardized) variables are given in the matrix A. When the principal components are obtained from a correlation matrix, A is the same as the matrix of unrotated factor loadings obtained for the principal components model for factor analysis.

Comments

Informational Errors

Type	Code	Description
3	1	Because NDF is zero or less, 100 degrees of freedom will be used.
3	2	One or more eigenvalues much less than zero are computed. The matrix COV is not nonnegative definite. In order to continue computations of STD and A, these eigenvalues are treated as zero.

Example

Principal components are computed for a nine‑variable matrix.

USE PRINC_INT

USE WRRRN_INT

IMPLICIT NONE

INTEGER ICOV, LDA, LDCOV, LDEVEC, NDF, NVAR

PARAMETER (ICOV=1, LDA=9, LDCOV=9, LDEVEC=9, NDF=100, NVAR=9)

REAL A(LDA,NVAR), COV(LDCOV,NVAR), EVAL(NVAR), &

EVEC(LDEVEC,NVAR), PCT(NVAR), STD(NVAR)

DATA COV/&

1.000, 0.523, 0.395, 0.471, 0.346, 0.426, 0.576, 0.434, 0.639, &

0.523, 1.000, 0.479, 0.506, 0.418, 0.462, 0.547, &

0.283, 0.645, 0.395, 0.479, 1.000, 0.355, 0.270, 0.254, &

0.452, 0.219, 0.504, 0.471, 0.506, 0.355, 1.000, 0.691, &

0.791, 0.443, 0.285, 0.505, 0.346, 0.418, 0.270, 0.691, &

1.000, 0.679, 0.383, 0.149, 0.409, 0.426, 0.462, 0.254, &

0.791, 0.679, 1.000, 0.372, 0.314, 0.472, 0.576, 0.547, &

0.452, 0.443, 0.383, 0.372, 1.000, 0.385, 0.680, 0.434, &

0.283, 0.219, 0.285, 0.149, 0.314, 0.385, 1.000, 0.470, &

0.639, 0.645, 0.504, 0.505, 0.409, 0.472, 0.680, 0.470, &

1.000/

CALL PRINC (NDF, COV, EVAL, ICOV=ICOV, PCT=PCT, STD=STD, &

EVEC=EVEC, A=A)

CALL WRRRN ('EVAL', EVAL, 1, NVAR, 1)

CALL WRRRN ('PCT', PCT, 1, NVAR, 1)

CALL WRRRN ('STD', STD, 1, NVAR, 1)

CALL WRRRN ('EVEC', EVEC)

CALL WRRRN ('A', A)

END

Output

EVAL

1 2 3 4 5 6 7 8 9

4.677 1.264 0.844 0.555 0.447 0.429 0.310 0.277 0.196

PCT

1 2 3 4 5 6 7 8 9

0.520 0.660 0.754 0.816 0.865 0.913 0.947 0.978 1.000

STD

1 2 3 4 5 6 7 8

0.6498 0.1771 0.0986 0.0879 0.0882 0.0890 0.0944 0.0994

0.1113

EVEC

1 2 3 4 5 6 7 8

1 0.3462 -0.2354 0.1386 -0.3317 -0.1088 0.7974 0.1735 -0.1240

2 0.3526 -0.1108 -0.2795 -0.2161 0.7664 -0.2002 0.1386 -0.3032

3 0.2754 -0.2697 -0.5585 0.6939 -0.1531 0.1511 0.0099 -0.0406

4 0.3664 0.4031 0.0406 0.1196 0.0017 0.1152 -0.4022 -0.1178

5 0.3144 0.5022 -0.0733 -0.0207 -0.2804 -0.1796 0.7295 0.0075

6 0.3455 0.4553 0.1825 0.1114 0.1202 0.0696 -0.3742 0.0925

7 0.3487 -0.2714 -0.0725 -0.3545 -0.5242 -0.4355 -0.2854 -0.3408

8 0.2407 -0.3159 0.7383 0.4329 0.0861 -0.1969 0.1862 -0.1623

9 0.3847 -0.2533 -0.0078 -0.1468 0.0459 -0.1498 -0.0251 0.8521

1 -0.0488

2 -0.0079

3 -0.0997

4 0.7060

5 0.0046

6 -0.6780

7 -0.1089

8 0.0505

9 0.1225

1 2 3 4 5 6 7 8

1 0.7487 -0.2646 0.1274 -0.2471 -0.0728 0.5224 0.0966 -0.0652

2 0.7625 -0.1245 -0.2568 -0.1610 0.5124 -0.1312 0.0772 -0.1596

3 0.5956 -0.3032 -0.5133 0.5170 -0.1024 0.0990 0.0055 -0.0214

4 0.7923 0.4532 0.0373 0.0891 0.0012 0.0755 -0.2240 -0.0620

5 0.6799 0.5646 -0.0674 -0.0154 -0.1875 -0.1177 0.4063 0.0039

6 0.7472 0.5119 0.1677 0.0830 0.0804 0.0456 -0.2084 0.0487

7 0.7542 -0.3051 -0.0666 -0.2641 -0.3505 -0.2853 -0.1589 -0.1794

8 0.5206 -0.3552 0.6784 0.3225 0.0576 -0.1290 0.1037 -0.0854

9 0.8319 -0.2848 -0.0072 -0.1094 0.0307 -0.0981 -0.0140 0.4485

1 -0.0216

2 -0.0035

3 -0.0442

4 0.3127

5 0.0021

6 -0.3003

7 -0.0482

8 0.0224

9 0.0543